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What are Geometric Series?

What are Geometric Series?. Sissa’s Reward. If you take a chess board and place one piece of rice on the first square, two on the second square, four on the third square, eight on the fourth square etc. How many pieces of rice would you have in total on the whole chess board?.

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What are Geometric Series?

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  1. What are Geometric Series?

  2. Sissa’s Reward If you take a chess board and place one piece of rice on the first square, two on the second square, four on the third square, eight on the fourth square etc. How many pieces of rice would you have in total on the whole chess board?

  3. Sn = a(1 - rn) 1 - r Consider n terms of a Geometric Sequence with first term a and common ratio r a, ar, ar2, ar3, ar4, … … … … …, arn-2, arn-1 Take the sum of the first n terms – call this 1 Sn = a + ar + ar2 + … … … + arn-1 Multiply the sum of the first n terms by r, the common ratio – call this 2 rSn = ar + ar2 + … … … + arn-1 + arn Subtract 2 from 1 Sn – rSn = a + ar + ar2 + … … … + arn-1 – (ar + ar2+ … + arn-1 + arn) Group like terms together Sn – rSn = a + (ar – ar) + (ar2 – ar2) + … … … + (arn-1 – arn-1) - arn Cancel out any terms that will Sn – rSn = a - arn Take out common factors Sn(1 – r) = a(1 – rn) Rearrange to find the formula for the sum of the first n terms

  4. Using Sn = a(1 - rn) 1 - r S64 = 1(1 - 264) 1 - 2 Sissa’s Reward Sequence is: 1, 2, 4, 8, 16, 32, … … … a = r = 2 1 Chessboard has 64 squares so n = 64 Sn =18 446 744 073 709 551 615

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